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Computación y Sistemas

On-line version ISSN 2007-9737Print version ISSN 1405-5546

Comp. y Sist. vol.13 n.2 Ciudad de México Oct./Dec. 2009




Associative Memory in a Continuous Metric Space: A Theoretical Foundation


Memoria Asociativa en un Espacio Métrico Continuo: Fundamentos Teóricos


Enrique Carlos Segura


Department of Computer Science, University of Buenos Aires Ciudad Universitaria, Pab.I, (1428) Buenos Aires, Argentina.


Article received on December 18, 2007
Accepted on August 07, 2008



We introduce a formal theoretical background, which includes theorems and their proofs, for a neural network model with associative memory and continuous topology, i.e. its processing units are elements of a continuous metric space and the state space is Euclidean and infinite dimensional. This approach is intended as a generalization of the previous ones due to Little and Hopfield. The main contribution of the present work is to integrate –and to provide a theoretical background that makes this integration consistent– two levels of continuity: i) continuous response processing units and ii) continuous topology of the neural system, obtaining a more biologically plausible model of associative memory. We present our analysis according to the following sequence of steps: general results concerning attractors and stationary solutions, including a variational approach for the derivation of the energy function; focus on the case of orthogonal memories, proving theorems on stability, size of attraction basins and spurious states; considerations on the problem of resolution, analyzing the more general case of memories that are not orthogonal, and with possible modifications to the synaptic operator; getting back to discrete models, i. e. considering new viewpoints arising from the present continuous approach and examine which of the new results are also valid for the discrete models; discussion about the generalization of the non deterministic, finite temperature dynamics.

Keywords: associative memory, continuous metric space, dynamical systems, Hopfield model, stability, Glauber dynamics, continuous topology.



Presentamos bases teóricas formales, incluyendo teoremas y sus demostraciones, para un modelo de red neuronal con memoria asociativa y topología continua, i. e. sus unidades de procesamiento son elementos de un espacio métrico continuo y el espacio de estados es euclidiano y de dimensión infinita. El enfoque es concebido como una generalización de los precedentes debidos a Little y Hopfield. La principal contribución del presente trabajo es integrar –y proveer fundamentos teóricos que den consistencia a tal integración– dos niveles de continuidad: i) unidades de proceso de respuesta continua y ii) topología continua del sistema neuronal, obteniendo de esta manera un modelo mas biológicamente plausible de memoria asociativa. Nuestro análisis es presentado de acuerdo con la siguiente secuencia de pasos: resultados generales sobre atractores y soluciones estacionarias, que incluyen un enfoque variacional para derivar la función de energía; estudio detallado del caso de memorias ortogonales, demostrando teoremas sobre estabilidad, tamaño de cuencas de atracción y estados espurios; consideraciones sobre el problema de la resolución, analizando el caso más general de memorias no ortogonales, y con modificaciones posibles al operador sináptico; retorno a los modelos discretos, i.e. consideración de nuevos puntos de vista que surgen del presente esquema, y de cuales de los nuevos resultados son también válidos para los modelos discretos; discusión sobre la generalización de la dinámica no deterministica a temperatura finita.

Palabras clave: memoria asociativa, espacio métrico continuo, sistema dinámico, modelo de Hopfield, dinámica de Glauber, topología continua, estabilidad.





This work was partially supported by grants X166 from the University of Buenos Aires and 26001 from the National Agency for Scientific Research, Argentina.



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